Solve Quadratic Equations By Factoring: A Step-by-Step Guide
Hey guys! Today, we're diving into the world of quadratic equations and learning how to solve them using a nifty technique called factoring. We'll tackle a specific equation, 5x = 33x + 56, and break down each step so you can confidently solve similar problems. Plus, we'll explore how to check our answers using substitution and graphing utilities. So, buckle up and let's get started!
Understanding Quadratic Equations
Before we jump into factoring, let's quickly recap what quadratic equations are. A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (usually 'x') is 2. The standard form of a quadratic equation is:
ax² + bx + c = 0
Where 'a', 'b', and 'c' are constants, and 'a' is not equal to 0. These equations pop up everywhere in math and real-world applications, from calculating projectile motion to designing bridges. That's why mastering how to solve them is super important. In our case, the initial equation 5x = 33x + 56 doesn't look like the standard form, but don't worry, we'll get there!
Step 1: Rewriting the Equation in Standard Form
The first thing we need to do is rearrange our equation, 5x = 33x + 56, into the standard quadratic form (ax² + bx + c = 0). This involves moving all the terms to one side of the equation, leaving zero on the other side. To do this, let's subtract 33x and 56 from both sides of the equation:
5x - 33x - 56 = 33x + 56 - 33x - 56
This simplifies to:
-28x - 56 = 0
Oops! It seems there was a slight misunderstanding in the original equation. It appears we're missing an x² term, which means this isn't a quadratic equation after all but a linear equation. My bad! Let’s assume the original equation was a typo and should have been 5x² = 33x + 56. In that case, after moving the terms, we would get:
5x² - 33x - 56 = 0
Now, this looks much more like a quadratic equation we can work with! Identifying the correct form is crucial, guys, because it dictates our next steps. We have 'a' as 5, 'b' as -33, and 'c' as -56. This sets us up perfectly for the next stage: factoring.
Step 2: Factoring the Quadratic Equation
Now comes the fun part: factoring. Factoring is like reverse multiplication; we're trying to find two binomials that, when multiplied together, give us our quadratic equation. There are different techniques for factoring, but we'll focus on the trial-and-error method here, which is super useful for equations where 'a' isn't 1. So, we need to find two binomials in the form of (px + q)(rx + s) such that:
p * r = a = 5p * s + q * r = b = -33q * s = c = -56
This might sound intimidating, but let's break it down. We need two numbers that multiply to 5. Since 5 is a prime number, the options are pretty straightforward: 1 and 5. So, our binomials will look something like (5x + q)(x + s). Next, we need to find two numbers, q and s, that multiply to -56 and satisfy the middle term condition. The factors of 56 are: 1 and 56, 2 and 28, 4 and 14, 7 and 8. Since we need a negative product, one of the numbers must be negative. After some trial and error, we find that 7 and -8 (or -7 and 8) might work. Let’s try (5x + 7)(x - 8):
Expanding this, we get:
5x² - 40x + 7x - 56 = 5x² - 33x - 56
Bingo! This matches our quadratic equation. So, the factored form of 5x² - 33x - 56 is (5x + 7)(x - 8). See? It's like detective work, piecing together the clues. Factoring is a critical skill in solving quadratic equations, and practice makes perfect, so don't worry if it seems tricky at first. Keep at it!
Step 3: Solving for x
Now that we have our factored equation, (5x + 7)(x - 8) = 0, we can use the Zero Product Property. This property states that if the product of two factors is zero, then at least one of the factors must be zero. In other words, if A * B = 0, then either A = 0 or B = 0 (or both). Applying this to our equation, we set each factor equal to zero:
5x + 7 = 0x - 8 = 0
Now, we solve each equation separately. For the first equation, 5x + 7 = 0, we subtract 7 from both sides:
5x = -7
Then, we divide by 5:
x = -7/5
For the second equation, x - 8 = 0, we simply add 8 to both sides:
x = 8
So, we have two solutions: x = -7/5 and x = 8. These are the values of x that make our original quadratic equation true. Pretty cool, huh? But we're not done yet! We need to check our answers to make sure they're correct.
Step 4: Checking the Solutions by Substitution
To check our solutions, we substitute each value of x back into the original equation, 5x² = 33x + 56, and see if it holds true. Let's start with x = -7/5:
5(-7/5)² = 33(-7/5) + 56
5(49/25) = -231/5 + 56
49/5 = -231/5 + 280/5
49/5 = 49/5
It checks out! Now let's try x = 8:
5(8)² = 33(8) + 56
5(64) = 264 + 56
320 = 320
This solution also works! So, we've confirmed that both x = -7/5 and x = 8 are correct solutions to our quadratic equation. Substitution is a powerful way to verify your answers, and it's always a good habit to get into. But there's another way we can check our work: using a graphing utility.
Step 5: Checking the Solutions Using a Graphing Utility
Another fantastic way to check our solutions is by using a graphing utility, like a graphing calculator or online tools like Desmos or GeoGebra. We can graph the quadratic equation and identify the x-intercepts, which are the points where the graph crosses the x-axis. These x-intercepts represent the solutions to the equation. To do this, we graph the function:
y = 5x² - 33x - 56
The x-intercepts of the graph will be the values of x for which y = 0, which are exactly the solutions to our equation. If you graph this function, you'll see that it crosses the x-axis at approximately x = -1.4 (which is -7/5) and x = 8. This confirms our solutions from the factoring method! Graphing utilities provide a visual confirmation of our algebraic solutions, making them an invaluable tool for checking our work and understanding the behavior of quadratic equations.
Conclusion: Mastering Factoring for Quadratic Equations
Alright, guys! We've walked through the entire process of solving the quadratic equation 5x² = 33x + 56 by factoring. We covered rewriting the equation in standard form, factoring the quadratic expression, using the Zero Product Property to find the solutions, and checking our answers using both substitution and a graphing utility. Factoring can seem a bit like a puzzle at first, but with practice, it becomes a powerful tool in your mathematical arsenal. Remember, the key is to break down the problem into manageable steps and don't be afraid to experiment with different factors. And always, always check your work! Whether it's through substitution or graphing, verifying your solutions is crucial for building confidence and ensuring accuracy. So, keep practicing, and you'll be a factoring pro in no time! Remember math can be fun and you can tackle any problem with the right approach. Keep up the great work and we'll see you in the next math adventure!
